exercise:0605c203bb: Difference between revisions

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(Created page with "<div class="d-none"><math> \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}</math></div> Let <math>b_k</math> denote the expected number of steps to reach <math>n</math> from <math>n-k</math>, in the process described in Exercise Exercise. <ul><li> Define <math>b_0 = 0</math>. Show that for <math>k > 0</mat...")
 
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\newcommand{\mathds}{\mathbb}</math></div> Let <math>b_k</math> denote the expected number of steps to
\newcommand{\mathds}{\mathbb}</math></div> Let <math>b_k</math> denote the expected number of steps to reach <math>n</math> from <math>n-k</math>, in the process described in [[exercise:Fd190e1214 |Exercise]].
reach
 
<math>n</math> from <math>n-k</math>, in the process described in Exercise [[exercise:Fd190e1214 |Exercise]].
<ul style="list-style-type:lower-alpha"><li>
<ul><li>
Define <math>b_0 = 0</math>.  Show that for <math>k  >  0</math>, we have
Define <math>b_0 = 0</math>.  Show that for <math>k  >  0</math>, we have



Latest revision as of 22:21, 15 June 2024

[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]b_k[/math] denote the expected number of steps to reach [math]n[/math] from [math]n-k[/math], in the process described in Exercise.

  • Define [math]b_0 = 0[/math]. Show that for [math]k \gt 0[/math], we have
    [[math]] b_k = 1 + \frac 1k \bigl(b_{k-1} + b_{k-2} + \cdots + b_0\bigr)\ . [[/math]]
  • Let
    [[math]] f(x) = b_0 + b_1 x + b_2 x^2 + \cdots\ . [[/math]]
    Using the recursion in part (a), show that [math]f(x)[/math] satisfies the differential equation
    [[math]] (1-x)^2 y' - (1-x) y - 1 = 0\ . [[/math]]
  • Show that the general solution of the differential equation in part (b) is
    [[math]] y = \frac{-\log(1-x)}{1-x} + \frac c{1-x}\ , [[/math]]
    where [math]c[/math] is a constant.
  • Use part (c) to show that
    [[math]] b_k = 1 + \frac 12 + \frac 13 + \cdots + \frac 1k\ . [[/math]]